# IJPAM: Volume 60, No. 4 (2010)

**SOLVING INVERSE PROBLEMS FOR THE HAMMERSTEIN**

INTEGRAL EQUATION AND ITS RANDOM ANALOG

USING THE ``COLLAGE METHOD'' FOR FIXED POINTS

INTEGRAL EQUATION AND ITS RANDOM ANALOG

USING THE ``COLLAGE METHOD'' FOR FIXED POINTS

Department of Mathematics and Statistics

College of Physical and Engineering Science

University of Guelph

50, Stone Road East, Guelph, Ontario, N1G 2W1, CANADA

e-mail: [email protected]

e-mail: [email protected]

Department of Economics, Business and Statistics

University of Milan

Milan, ITALY

e-mail: [email protected]

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario, N2L 3G1, CANADA

e-mail: [email protected]

**Abstract.**Many inverse problems in applied mathematics can be formulated as the approximation of a target element in a complete metric space by the fixed point of an appropriate contraction mapping
.
The method of *collage coding* seeks to solve this problem by finding a contraction mapping that minimizes the
so-called *collage distance* . In this paper, we develop a collage coding framework for inverse problems
involving deterministic or random Hammerstein integral operators. Such operators are used to model image blurring. We
illustrate the method with examples.

**Received: **March 6, 2010

**AMS Subject Classification: **45Q05, 45R05, 60H25

**Key Words and Phrases: **Hammerstein integral equations, inverse problems, fixed point equations, random fixed point equations, collage theorem

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 60

**Issue:** 4